2)Felix landed 9 minutes and 3 seconds after jumping. How much quicker would an object in freefall all the way down (ie without a parachute) land?
->First, we need to assume few things. Due to these assumptions, terminal velocity of Felix is unlikely to be greater than the speed of sound. However, we should still get an approximate answer. I am also only using basic mathematics therefore, I am not going to use differential equations so that people without that knowledge can follow the steps.
We are going to start off with an assumption. The acceleration of Felix due to the gravity of earth is constant. This acceleration is going to be the mean of the acceleration at sea level and the acceleration at 39,000 m. The acceleration at 39,000 m can be worked out by using the following:
-> F = ma & F = (GMm)/r² ,where F is the force experience by a mass, M is the mass of the Earth, m is the mass of Felix, G is the Universal Gravitational constant and r is the distance between the centre of the mass of the Earth and the centre of the mass of Felix.
-> ma = (GMm)/r²
-> a = (GM)/r²
-> a = {(6.67*10^-11)(6.0*10^24)}
--------------------------
{(3.9*10^4 + 6.4*10^6)²} <- (Since r is the distance between the
centre of the mass of the Earth and
the centre of the mass of Felix, we
need to add the radius of the Earth
and the distance between the surface
of the Earth and Felix to get the
separation.)
-> a at 39,000 m = 9.65 ms^-2 (approx.)
-> Therefore, average acceleration = (9.65 + 9.81)/2
= 9.73 ms^-2
We are going to take this value, 9.73 ms^-2, and treat it as a constant acceleration to make our mathematics easier.
Assumption 1: Constant acceleration due to gravity = 9.73 ms^-2
Next, we need to know how to find the terminal velocity of an object. This is given by the following equation:
-> V² = {(2*m*g)/(P*A*Cd)} ,where V is the terminal velocity of the object, m is the mass of the object, g is the acceleration due to gravity, P is the density of the fluid through which the object is travelling through, A is the projected area of front of the object and Cd is the drag coefficient of the object.
Some of these can only be obtained through research such as the mass of Felix and the drag coefficient of Felix. Others, like the density of the air through which the Felix is travelling through must be worked out separately. If we were to make a list of things we need to research and things we need to calculate, it would look something like this:
-> Things we need to research : Drag coefficient of Felix, mass of Felix
-> Things we need to calculate : Fluid density, Projected area of the front
of Felix, Terminal velocity
Things we need to research :
Drag Coefficient :
Drag coefficient of a normal person is about 1-1.3. Since Felix carrying all those gears and chest packs, let's assume that his drag coefficient is at the higher end (1.3).
Assumption 2: Drag Coefficient = 1.3
Mass of Felix :
In the official website of Red Bull, they have stated that including equipments, Felix had a mass of 260 pounds which is about 120 kg.
Things we need to calculate:
Projected area of the front of Felix:
When upright, a normal body takes around 0.1 m² but since he was tumbling a little bit, we can assume the projected area of the front of Felix to be about 0.15 m²
Assumption 3: Projected area in front of Felix = 0.15 m² (approx.)
Fluid Density:
To calculate this, we need a bit more mathematics. Fluid density can be worked out by using the following equation:
-> P = p/(RT) ,where P is the fluid density, p is the absolute pressure, R is the specific gas constant and T is the Absolute temperature.
Further research on the internet reveals that the absolute pressure at 39,000 is only about 0.33% of the absolute pressure at sea level (101000 pa). Given this information, We can work out the absolute pressure at 39,000 m.
-> p = (0.33*101000)/100
-> p = 333.3 Pa (approx.)
The specific gas constant is 8.31 JK^-1 and further research tells us that at 39,000 m, the temperature was -14 degrees Fahrenheit. However, we need the Absolute temperature therefore the Absolute temperature at 39,000 m is 247.59 Kelvin.
We now, have everything we need to work out P.
-> P = p/(RT)
-> P = 333.3/(8.31*247.59)
-> P = 0.162 kgm^-3 (approx.)
With this last piece of data, we can now finally work out the terminal velocity.
-> V² = {(2*m*g)/(P*A*Cd)}
-> V² = {(2*120*9.73)}/(0.162*0.15*1.3)
-> V² = 7.4*10^4 m²s^-4 (approx.)
-> V = 270 ms^-1
This means that the terminal velocity of Felix, given that our assumptions are correct, is about 270 ms^-1 which is below the speed of sound at 39,000 m (which is approximately 316 ms^-1).
Since we now know the terminal velocity, we can use the equations of motion with constant acceleration to find time it would take Felix to reach the surface of the Earth if he had not opened his parachute. First of all, let's find out the time taken for Felix to reach his terminal velocity of 270 ms^-1 given that the acceleration is 9.73 ms^-2.
-> v = u + at ,where v is the final velocity, u is the initial velocity, a is the acceleration due to gravity and t is the time taken.
-> t = (v - u)a
-> t = 270/9.73
-> t = 28 seconds (approx.)
Now let's find out the distance travelled in these 28 seconds so that we can work out how high he is from the ground.
-> s = ut + (at²)/2 ,where s is the displacement of the object from it's initial position.
-> s = 0 + (9.73*28²)/2
-> s = 3,800 m (approx.)
Since he has travelled 3,800 m in 28 second, we can work out the distance left between Felix and the ground.
-> d = 39,000 - 3,800 , where d is the distance between Felix and the ground
-> d = 35,200 m
Now that Felix has reached his terminal velocity, we are going to assume that his velocity does not change for the rest of the fall. This means that we will ignore the increase is air resistance and air density increases with decrease in height. This will make the mathematics a bit easier because if we took into account the increase in air resistance, we would have to use differential equations.
Assumption 4: Terminal velocity does not change.
Since he is travelling at 270 ms^-1 and that the distance between Felix and the ground is 35,200 m, We can do a simple calculation to workout the time taken for Felix to reach the ground.
-> v = s/t , where v is the velocity of an object, s is the displacement of the object and t is the time taken.
-> t = s/v
-> t = 35,200/270
-> t = 130 s
We can now add the time taken by Felix to reach his terminal velocity and the time taken to reach the ground after Felix reached his terminal velocity to get the total time of free fall.
-> Total time = 28 + 130
-> Total time = 158 s (approx.)
In conclusion, by my calculations (provided that there is no mistakes), it takes Felix about 2 minutes and 38 seconds to reach the surface of the Earth if he fell from 39,000 m without opening his parachute.